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%I A358550 #7 Nov 22 2022 11:57:59
%S A358550 1,2,2,3,2,3,3,3,4,2,3,3,3,4,3,3,3,3,4,4,4,4,5,2,3,3,3,4,3,3,3,3,4,4,
%T A358550 4,4,5,3,3,3,3,4,3,3,3,3,4,4,4,4,5,4,4,4,4,4,4,4,4,5,5,5,5,5,6,2,3,3,
%U A358550 3,4,3,3,3,3,4,4,4,4,5,3,3,3,3,4,3,3,3
%N A358550 Depth of the ordered rooted tree with binary encoding A014486(n).
%C A358550 The binary encoding of an ordered tree (A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.
%e A358550 The first few rooted trees in binary encoding are:
%e A358550     0: o
%e A358550     2: (o)
%e A358550    10: (oo)
%e A358550    12: ((o))
%e A358550    42: (ooo)
%e A358550    44: (o(o))
%e A358550    50: ((o)o)
%e A358550    52: ((oo))
%e A358550    56: (((o)))
%e A358550   170: (oooo)
%e A358550   172: (oo(o))
%e A358550   178: (o(o)o)
%e A358550   180: (o(oo))
%e A358550   184: (o((o)))
%t A358550 binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
%t A358550 bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
%t A358550 Table[Depth[bint[k]]-1,{k,Select[Range[0,1000],binbalQ]}]
%Y A358550 Positions of first appearances are A014137.
%Y A358550 Leaves of the ordered tree are counted by A057514, standard A358371.
%Y A358550 Branches of the ordered tree are counted by A057515.
%Y A358550 Edges of the ordered tree are counted by A072643.
%Y A358550 The Matula-Goebel number of the ordered tree is A127301.
%Y A358550 Positions of 2's are A155587, indices of A020988.
%Y A358550 The standard ranking of the ordered tree is A358523.
%Y A358550 Nodes of the ordered tree are counted by A358551, standard A358372.
%Y A358550 For standard instead of binary encoding we have A358379.
%Y A358550 A000108 counts ordered rooted trees, unordered A000081.
%Y A358550 A014486 lists all binary encodings.
%Y A358550 Cf. A001263, A057122, A358373, A358505, A358524.
%K A358550 nonn
%O A358550 1,2
%A A358550 _Gus Wiseman_, Nov 22 2022

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